Vexillary degeneracy loci classes in K-theory and algebraic cobordism
aa r X i v : . [ m a t h . AG ] J a n VEXILLARY DEGENERACY LOCI CLASSES IN K-THEORY ANDALGEBRAIC COBORDISM
THOMAS HUDSON AND TOMOO MATSUMURA
Abstract.
In this paper, we prove determinant formulas for the K -theory classes ofthe structure sheaves of degeneracy loci classes associated to vexillary permutations intype A . As a consequence we obtain determinant formulas for Lascoux–Sch¨utzenberger’sdouble Grothendieck polynomials associated to vexillary permutations. Furthermore, wegeneralize the determinant formula to algebraic cobordism. Introduction
Let E ⊂ · · · ⊂ E n − → H n − ։ · · · ։ H ։ H be a sequence of maps of vector bundlesover a nonsingular variety X where the subscripts indicate the rank. For a permutation w ∈ S n , we define the degeneracy locus X w in X by X w := X w ( E • → H • ) := { x ∈ X | rk( F q | x → H p | x ) ≤ r w ( p, q ) ∀ p, q } where r w ( p, q ) = ♯ { i ≤ p | w ( i ) ≤ q } . A permutation w ∈ S n is called vexillary if it avoidsthe pattern (2143), i.e. there is no a < b < c < d such that w ( b ) < w ( a ) < w ( d ) < w ( c ).The primary goal of this article is to prove determinantal formulas for the K -theory class[ O X w ] of the structure sheaf of X w , where w is a vexillary permutation (Theorem 4.1).Such formulas, which is written in terms of Segre classes, are later generalized to algebraiccobordism (Theorem 6.1). Our proof is based on our joint work [14] with Ikeda and Naruseand on [15], where determinantal formulas were obtained for Grassmannian permutations.Independently from this work, Anderson [1] also proved a determinant formula for [ O X w ]in terms of Chern classes with a similar method.Lascoux and Sch¨utzenberger ([22], [24]) introduced the double Grothendieck polyno-mial G w ( x | b ) to represent the K -theory classes of the structure sheaves of Schubert va-rieties and Fomin–Kirillov ([9], [8]) gave their combinatorial description in terms of pipedreams or rc graphs. In the case when w is vexillary, Knutson–Miller–Yong [20] described G w ( x | b ) in terms of flagged set-valued tableaux , unifying the work of Wachs [29] on flaggedtableaux, and Buch [5] on set-valued tableaux. If the codimension of X w coincides withthe length of w , then we know from the works of Fulton and Lascoux [12] and Buch [4]that [ O X w ] = G w ( x | b ) where we set x i = c (ker( H i → H i − )) and b i = c (( E i /E i − ) ∨ ).As a consequence, our formula also proves determinantal formulas of G w ( x | b ), generalizingthe Jacobi-Trudi type formulas of double Grothendieck polynomials associated to Grass-mann permutations obtained in [14] and [15]. Note that in [25], [17], [18], and [30], onecan find different determinantal formulas of Grothendieck polynomials for Grassmannian permutations as well as to certain vexillary permutations. Those formulas are in termsof complete/elementary symmetric polynomials while ours are in terms of Grothendieckpolynomials associated to one row partitions.Let us now recall the corresponding results for cohomology. Lascoux and Sch¨utzenbergerdefined double Schubert polynomials indexed by permutations in [21] and [23]. Fulton [10]showed that they give the cohomology classes of the degeneracy loci given by rank con-ditions associated to permutations. Lascoux and Sch¨utzenberger also introduced vexillarypermutations where the associated Schubert polynomials are given by generalized Schurdeterminants. In [10], Fulton gave a simple description of the degeneracy loci given byvexillary permutations (see also [3] and [2]) and proved the determinant formula for thecohomology classes of such degeneracy loci. For Grassmannian permutations (which areexamples of vexillary permutations), the corresponding (double) Schubert polynomials co-incide with the (factorial) Schur polynomials, and their determinant formula is known asthe Jacobi–Trudi formula or the determinant formulas of Kempf–Laksov [16] and Damon[7], which further specializes to Giambelli–Thom–Porteous formula.The paper is organized as follows. In Section 2, we review some of the basics on Segreclasses in connective K -theory. In Section 3, we recall the partition associated to a vexillarypermutation w and a flagging set of w , which corresponds to an (inflated) triple of type A in [2]. In Section 4, we describe our main theorem in terms of the partition and a flaggingset. The proof is based on the construction of a resolution using the data of the flaggingset and the computation of the class of X w as the pushforward of the class of the resolutionfollowing [14]. In Section 5, we describe the corresponding determinant formula for thevexillary double Grothendieck polynomials. In Section 6, we also take into considerationthe algebraic cobordism of Levine–Morel [26] and describe the corresponding classes as aninfinite linear combination of Schur determinants whose coefficients are given by a certainpower series defined by means of the universal formal group law.In this paper, all schemes and varieties are assumed to be quasi-projective over analgebraically closed field k of characteristic zero, unless otherwise stated.2. Connective K -theory and Segre classes Connective K -theory, denoted by CK ∗ , is an example of oriented cohomology theorybuilt out of the algebraic cobordism of Levine and Morel. For the detailed constructionwe refer the reader to [6], [13], and [26]. Connective K -theory assigns to a smooth variety X a commutative graded algebra CK ∗ ( X ) over the coefficient ring CK ∗ ( pt ). Here the ring CK ∗ ( pt ) is isomorphic to the polynomial ring Z [ β ] by setting β to be ( −
1) times the class ofdegree − P → pt . The Z [ β ]-algebra CK ∗ ( X ) specializes to the Chow ring CH ∗ ( X ) and the Grothendieck ring K ( X ) by respectively setting β equal to 0 and −
1. Forany closed equidimensional subvariety Y of X , there exists an associated fundamental class[ Y ] CK ∗ in CK ∗ ( X ) which specializes to the class [ Y ] in CH ∗ ( X ) and also to the class of the EXILLARY DEGENERACY LOCI CLASSES IN K-THEORY AND ALGEBRAIC COBORDISM 3 structure sheaf O Y of Y in K ( X ). In the rest of the paper, we denote the fundamentalclass of Y in CK ∗ ( X ) by [ Y ] instead of [ Y ] CK ∗ .As a feature of any oriented cohomology theory, connective K -theory admits a theoryof Chern classes. For a line bundle L , its first Chern class c ( L ) corresponds to the usualChern class in CH ∗ ( X ) and to the class 1 − [ L ∨ ] in K ( X ) under the specialization β = 0and β = − L and L over X , their 1st Chern classes c ( L i ) ∈ CK ( X ) satisfy the following equality: c ( L ⊗ L ) = c ( L ) + c ( L ) + βc ( L ) c ( L ) . It follows that c ( L ∨ ) = − c ( L )1+ βc ( L ) . For a variable u , we denote ¯ u := − u βu . The reader shouldbe aware that our sign convention for β is opposite to the one used in the references [6],[13], [26]. For computations it is convenient to combine the Chern classes into a Chernpolynomial c ( E ; u ) := P ei =0 c i ( E ) u i . For each virtual bundle E − F as an element of K ( X ),one has the relative Chern classes c i ( E − F ) defined by c ( E − F ; u ) := c ( E ; u ) /c ( F ; u ). Definition 2.1.
We define the
Segre classes for a virtual bundle E − F by the following: S ( E − F ; u ) := X m ∈ Z S m ( E − F ) u m = 11 + βu − c ( E − F ; β ) c ( E − F ; − u ) . (2.1)Consequently the Segre classes satisfy the following identities ( cf. [14], [15]): S m ( E − F ) = rk F X p =0 ( − p c p ( F ) S m − p ( E ) 1 c ( F ; β ) (2.2)= rk F X p =0 c p ( F ∨ ) p X q =0 (cid:18) pq (cid:19) β q S m − p + q ( E ) . (2.3)The following proposition was obtained in [14] based on the results in [4] and [28]. Proposition 2.2.
Let π : P ∗ ( E ) → X be the dual projective bundle of a rank e vectorbundle E → X and let τ be the first Chern class of its tautological quotient line bundle Q .Let F be a vector bundle over X of rank f and denote its pullback to P ∗ ( E ) also by F . Wehave π ∗ ( τ s c f ( Q ⊗ F ∨ )) = S s + f − e +1 ( E − F ) . Remark 2.3.
When E is a line bundle, we have c ( E ) s c f ( E ⊗ F ∨ ) = S f + s ( E − F ) byregarding E as the tautological quotient line bundle of P ∗ ( E ).3. Vexillary permutation
Let S n be a permutation group of { , . . . , n } and w ∈ S n . The rank function of w isdefined by r w ( p, q ) := ♯ { i ≤ p | w ( i ) ≤ q } . The diagram of w , denoted by D ( w ), is defined by D ( w ) := { ( p, q ) ∈ { , . . . , n } × { , . . . , n } | π ( p ) > q, and π − ( q ) > p } . THOMAS HUDSON AND TOMOO MATSUMURA
We call an element of the grid { , . . . , n }×{ , . . . , n } a box. Two elements ( p, q ) and ( p ′ , q ′ )in D ( w ) are said to be connected we can find a sequence of adjacent boxes connecting themthat are contained in D ( w ). The essential set E ss ( w ) of w is the subset of D ( w ) given by E ss ( w ) := { ( p, q ) | p, q ≤ n − , w ( p ) > q, w ( p + 1) ≤ q, w − ( q ) > p, and w − ( q + 1) ≤ p } . A permutation w ∈ S n is called vexillary if it avoids the pattern (2143), i.e. there is no a < b < c < d such that w ( b ) < w ( a ) < w ( d ) < w ( c ). A characterization of vexillarypermutations in terms of the essential sets has been given by Fulton [10, Proposition 9.6].In particular, w is vexillary if and only if the boxes in E ss ( w ) are placed along the directiongoing from north-east to south-west, i.e. for any ( p, q ) , ( p ′ , q ′ ) ∈ E ss ( w ), p ≥ p ′ implies q ≤ q ′ .A partition λ of length r is a non-increasing sequence of r positive integers ( λ , . . . , λ r )with the convention that λ i = 0 for i > r . We identify a partition λ with its Young diagram { ( i, j ) | ≤ i ≤ r, ≤ j ≤ λ i } in English notation. For each vexillary permutation w ∈ S n ,one can assign a partition λ ( w ): let the number of boxes ( i, i + k ) in the k -th diagonal ofthe Young diagram of λ ( w ) be equal to the number of boxes in the k -th diagonal of D ( w )for each k (see [20, 19]). This defines a bijection φ from D ( w ) to λ ( w ), which takes the j -th box in the k -th diagonal of D ( w ) to the j -th box in the k -th diagonal of λ ( w ). Inother words, we set φ ( p, q ) = ( p − r w ( p, q ) , q − r w ( p, q )) for each ( p, q ) ∈ D ( w ). Under thisbijection φ , the boxes in E ss ( w ) correspond bijectively to the south-east corners of λ ( w ). Definition 3.1.
Let w ∈ S n be a vexillary permutation. A subset f ( w ) = { ( p i , q i ) , i =1 , . . . , d } of { , . . . , n } × { , . . . , n } is called a flagging set for w if it satisfies(i) p ≤ p ≤ · · · ≤ p d , q ≥ q ≥ · · · ≥ q d ;(ii) The set f ( w ) contains E ss ( w );(iii) p i − r i = i for i = 1 , . . . , d where r i := r w ( p i , q i ).Although you can find the following claim in [3, 2], we give a proof for completeness ( cf. [10, 27]). Lemma 3.2. If f ( w ) = { ( p i , q i ) , i = 1 , . . . , d } is a flagging set of a vexillary permutation w ∈ S n , then the partition λ ( w ) is given by λ i = q i − p i + i for each i = 1 , . . . , d .Proof. If ( p i , q i ) is in E ss ( w ), it maps to a southeast corner of λ under φ and hence (iii)implies λ i = q i − p i + i . Suppose ( p i , q i ) is the first element that is contained in E ss ( w )so that there is no box in the diagram to the east of ( p i , q i ). We see that the condition(iii) implies that q j − p j = λ j − j for all j = 1 , . . . , i −
1. Next let ( p i , q i ) and ( p j , q j ) beconsecutive elements of the flagging set which are contained in E ss ( w ) with i > j . Considerthe ( p i − p j + 1) × ( q j − q i + 1) rectangle R in { , ..., n } × { , ..., n } with ( p j , q j ) as thenortheast corner and ( p i , q i ) as the southwest corner. For i > k > j , ( p k , q k ) must be in R .Since w is vexillary, E ss ( w ) ∩ R consists of only ( p i , q i ) and ( p j , q j ). This implies that anelement of R is in D ( w ) only if it is on the leftmost column or on the top row. We canalso observe that the northwest corner of R is contained in D ( w ) and connected to ( p i , q i ) EXILLARY DEGENERACY LOCI CLASSES IN K-THEORY AND ALGEBRAIC COBORDISM 5 or ( p j , q j ). Moreover we see that the number of connected components of D ( w ) ∩ R is atmost two. Thus we have one of the following three cases: (1) D ( w ) ∩ R is a hook shape,(2) D ( w ) ∩ R contains the top row, or (3) D ( w ) ∩ R contains the leftmost column. In eachcase, we see by the condition (iii) that, for i > k > j , ( p k , q k ) and φ − ( k, λ k ) are on thesame diagonal, i.e. λ k − k = q k − p k . This proves the claim for r ≥ i where r is the lengthof λ ( w ). Suppose that i > r . Then ( p i , q i ) must be in the southwest ( n − p r + 1) × q r rectangle R ′ containing ( p r , q r ) in { , . . . , n } × { , . . . , n } . Since q r is the maximum rowindex of D ( w ), we can see that p − r w ( p, q ) = i if and only if q − p = − i . Hence we have q i − p i + i = 0. This completes the proof. (cid:3) Determinant formula of X w ( E • → H • )Let E ⊂ · · · ⊂ E n − φ → H n − ։ · · · ։ H ։ H be a full flag of vector bundles on X followed by a map to a dual full flag. The subscripts indicate the rank. For each w ∈ S n ,we define the degeneracy locus X w in X by X w := X w ( E • → H • ) := { x ∈ X | rk( E q | x → H p | x ) ≤ r w ( p, q ) ∀ p, q } . By Proposition 4.2 [10], it suffices to consider the rank conditions for ( p, q ) ∈ E ss ( w ). Inwhat follows, we assume that the bundles and maps are sufficiently generic so that thecodimension of X w is the length ℓ ( w ) of w . Suppose that w is vexillary and choose aflagging set { ( p i , q i ) , i = 1 , . . . , d } of w , then we can write X w = { x ∈ X | rk( E q i | x → H p i | x ) ≤ r w ( p i , q i ) ∀ i = 1 , . . . , d } . The following is our main theorem.
Theorem 4.1.
Let w ∈ S n be a vexillary permutation and f ( w ) = { ( p i , q i ); i = 1 , . . . , d } an arbitrary flagging set of w . Let A [ i ] m := A [ i ] m ( f ( w )) := S m ( H p i − E q i ) for each m ∈ Z .Then the class of the degeneracy locus X w in CK ∗ ( X ) is given by [ X w ] = det ∞ X s =0 (cid:18) i − ds (cid:19) β s A [ i ] λ i + j − i + s ! ≤ i,j ≤ d (4.1) and also by [ X w ] = det ∞ X s =0 (cid:18) i − js (cid:19) β s A [ i ] λ i + j − i + s ! ≤ i,j ≤ d , (4.2) where λ ( w ) = ( λ , . . . , λ d ) . Remark 4.2.
Independently from our work, Anderson [1] obtained a determinant formulaof [ X w ] in terms of Chern classes, which is equivalent to (4.2) via writing Segre classes interms of Chern classes. In particular, his formula also gives a formula which depends onlyon the data of the essential set of w . THOMAS HUDSON AND TOMOO MATSUMURA
Resolution of X w . Consider the dual flag bundle π : F l ∗ ( H p ) → X whose fiber at x consists of dual flags D d | x ։ · · · ։ D | x with dim D i | x = i and the commutative diagramof surjective maps H p d | x / / (cid:15) (cid:15) H p d − | x / / (cid:15) (cid:15) · · · / / H p | x / / (cid:15) (cid:15) H p | x (cid:15) (cid:15) D d | x / / D d − | x / / · · · / / D | x / / D | x . Let D i be the tautological bundle of rank i . It can be constructed as the following tower F l ∗ ( H p ) := P ∗ (ker( H p d → D d − )) → · · · → P ∗ (ker( H p → D )) → P ∗ ( H p ) → X. We can regard ker( D i → D i − ) as the tautological quotient line bundle of P ∗ (ker( H p i → D i − )) where we set D = 0. Definition 4.3.
For each flagging set f ( w ) = { ( p i , q i ) , i = 1 , . . . , d } , define a sequence ofsubvarieties Y d ⊂ · · · ⊂ Y ⊂ F l ∗ ( H p ) by Y j := { ( x, D • | x ) ∈ F l ∗ ( H p ) | rk( E q i | x → D i | x ) = 0 , ≤ i ≤ j } . Denote Y d by Y f ( w ) .It is well known that π maps Y f ( w ) to X w birationally and that X w has at worst rationalsingularities, therefore π ∗ [ Y f ( w ) ] = [ X w ] in CK ∗ ( X ). Lemma 4.4. In CK ∗ ( F l ∗ ( H p )) , we have [ Y f ( w ) ] = r Y i =1 c q i (ker( D i → D i − ) ⊗ E ∨ q i ) . Proof.
By definition, there is an obvious bundle map E q i → ker( D j → D j − ) over Y j − andits zero locus in Y j − coincides with Y j . Thus from the standard fact ( cf. [14, Lemma 2.2])we obtain [ Y j ] = c q j (ker( D j → D j − ) ⊗ E ∨ q j )in CK ∗ ( Y j − ). The claim follows from the projection formula. (cid:3) Pushforward formula.
Set R = CK ∗ ( X ). This is a graded algebra over Z [ β ]. Let t , . . . , t d be indeterminates of degree 1. We use the multi-index notation t s := t s · · · t s d d for s = ( s , . . . , s d ) ∈ Z d . A formal Laurent series f ( t , . . . , t d ) = X s ∈ Z d a s t s is homogeneous of degree m ∈ Z if a s is zero unless a s ∈ R m −| s | with | s | = P di =1 s i . Letsupp f = { s ∈ Z d | a s = 0 } . For each m ∈ Z , define L Rm to be the space of all formalLaurent series of homogeneous degree m such that there exists n ∈ Z d such that n + supp f is contained in the cone in Z d defined by s ≥ , s + s ≥ , · · · , s + · · · + s d ≥
0. Then L R := L m ∈ Z L Rm is a graded ring over R with the obvious product. For each i = 1 , . . . , d , EXILLARY DEGENERACY LOCI CLASSES IN K-THEORY AND ALGEBRAIC COBORDISM 7 let L R,i be the R -subring of L R consisting of series that do not contain any negative powersof t , . . . , t i − . In particular, L R, = L R . A series f ( t , . . . , t d ) is a power series if it doesn’tcontain any negative powers of t , . . . , t d . Let R [[ t , . . . , t d ]] m denote the set of all powerseries in t , . . . , t d of degree m ∈ Z . We set R [[ t , . . . , t d ]] gr := L m ∈ Z R [[ t , . . . , t d ]] m . Definition 4.5.
For each j = 1 , . . . , d , define a graded R -module homomorphism φ j : L R,j → CK ∗ ( P ∗ (ker( H p j − → D j − ))by setting φ j ( t s · · · t s d d ) = τ s · · · τ s j − j − A [ j ] s j · · · A [ d ] s d where τ i := c (ker( D i → D i − )). Lemma 4.6.
Let π j : P ∗ (ker( H p j → D j − )) → P ∗ (ker( H p j − → D j − )) . We have π j ∗ ( τ s c q j (ker( D j → D j − ) ⊗ E ∨ q j )) = φ j t λ j j Y ≤ i Proposition 4.7. We have π ∗ [ Y f ( w ) ] = φ d Y j =1 t λ j j · Y ≤ i By the formula of the Vadermonde determinant, we have d Y j =1 t λ j j · Y ≤ i Vexillary double Grothendieck polynomials For a given partition λ , a flagging f of λ is a sequence of natural numbers ( f , . . . , f r )where r is the length of λ . A flagged set-valued tableaux in shape λ with flagging f isa set-valued tableaux of shape λ such that the numbers used in the i -th row are at most f i . Let f ( w ) = { ( p i , q i ) , i = 1 , . . . , r } be a flagging set of w and consider the flagging f w with ( f w ) i := p i . Let FSVT ( w ) be the set of all flagged set-valued tableaux of w withthe flagging f w . It is worth remaking that the set FSVT ( w ) is the same for any choice offlagging f such that p ′ ≤ f i ≤ p where p ′ is the row index of φ − ( i, λ i ) and p is the rowindex of the preimage under φ of the south-east corner of λ whose column contains ( i, λ i ).Let x = ( x , x , . . . ) and b = ( b , b , . . . ) be sets of infinitely many indeterminants. In[20] (see also [19]), Knutson–Miller–Yong described the double Grothendick polynomialsof Lascoux–Sch¨utzenberger [24] associated to a vexillary permutation w as a generatingfunction of flagged set-valued tableaux. Namely we have G w ( x | b ) = X τ ∈ FSVT ( w ) ( − | τ |−| λ ( w ) | Y e ∈ τ ( x val ( e ) ⊕ y val ( e )+ c ( e ) − r ( e ) ) , where val ( e ) is the numerical value of e and c ( e ) and r ( e ) are the column and row indices of e respectively. Note that we replaced 1 − x i and 1 − y i − in [20] by x i and by b i respectively.We know from the work of Fulton–Lascoux [12] and Buch [4] that[ X w ] = G w ( x | b ) , (5.1)where we set x i = c (ker( H i → H i − )) and b i = c (( E i /E i − ) ∨ ).For nonnegative integers p, q and an integer m , we define G [ p,q ] m ( x | b ) by X m ∈ Z G [ p,q ] m ( x | b ) u m = 11 + βu − Y ≤ i ≤ p βx i − x i u Y ≤ j ≤ q (1 + b j ( β + u )) . We see from Definition 2.1 that G [ p,q ] m ( x | b ) = S m ( E − F ) for vector bundles E and F ofrank p and q with Chern roots { x i } and { ¯ b i } respectively. Consequently we can derive from EXILLARY DEGENERACY LOCI CLASSES IN K-THEORY AND ALGEBRAIC COBORDISM 9 Theorem 4.1 the following Jacobi-Trudi type formulas of vexillary double Grothendieckpolynomials. Theorem 5.1. Let w ∈ S n be a vexillary permutation. For an arbitrary flagging set { ( p i , q i ) , i = 1 , . . . , d } of w , we have G w ( x | b ) = det ∞ X s =0 (cid:18) i − ds (cid:19) β s G [ p i ,q i ] λ i + j − i + s ( x | b ) ! ≤ i,j ≤ d , and also G w ( x | b ) = det ∞ X s =0 (cid:18) i − js (cid:19) β s G [ p i ,q i ] λ i + j − i + s ( x | b ) ! ≤ i,j ≤ d , where λ ( w ) = ( λ , . . . , λ d ) . Generalization to algebraic cobordism In this section, we discuss the generalization of Theorem 4.1 to the algebraic cobordismof Levine–Morel [26]. Let Ω ∗ ( X ) denote the algebraic cobordism of X .First we recall, from [15], the formulas for the Segre classes in algebraic cobordism. Let F Ω ( u, v ) be the formal group law of algebraic cobordism, the universal one defined overthe Lazard ring L . We denote the formal inverse of x by χ ( x ). Let P ( z, x ) be the uniquepower series in x and z defined by F Ω ( z, χ ( x )) = ( z − x ) P ( z, x ). Let E and F denotevector bundles on X of rank e and f respectively. For each integer s ≥ 0, we define theclass w − s ( E ) in Ω − s ( X ) by the following generating function w ( E ; u ) := ∞ X s =0 w − s ( E ) u − s := e Y q =1 P ( u − , x q ) , where x q for q = 1 , . . . , e are Chern roots of E . Moreoever, let P ( u ) := ∞ X i =0 [ P i ] u − i , where [ P i ] is the class of the projective space P i of degree − i in L . We can define the relativeSegre classes of a virtual bundle E − F in Ω ∗ ( X ) by the following generating function: S ( E − F ; u ) := X k ∈ Z S k ( E − F ) u k := P ( u ) c ( E − F ; − u ) w ( E − F ; u ) . (6.1)By Theorem 3.9 in [15], we can express the relative Segre classes by the pushforwardof Chern classes along a projective bundle as follows. Let π : P ∗ ( E ) → X be the dualprojective bundle of E , Q its tautological quotient line bundle, and τ := c ( Q ). We have π ∗ ( τ s c f ( Q ⊗ F ∨ )) = S f − e +1+ s ( E − F ) . (6.2)We see from this that the Segre class S m ( E ) for a vector bundle E is the natural general-ization of the one given by Fulton in [11]. Now we describe the main theorem in this section. Let f ( w ) = { ( p i , q i ) , i = 1 , . . . , d } bea flagging set of a vexillary permutation w ∈ S n . We define Y f ( w ) as in Definition 4.3 andalso let A [ i ] m := A [ i ] m ( f ( w )) := S m ( H p i − E q i ) in Ω ∗ ( X ) as before. If we set Y ≤ i For each flagging set f ( w ) = { ( p i , q i ) | i = 1 , . . . , d } of a vexillary permu-tation w ∈ S n , we have [ Y f ( w ) → X ] = X s =( s ,...,s d ) ∈ Z d ≥ a s ∆ λ ( w )+ s ( A [1] , . . . , A [ d ] ) . Proof. The proof is similar to the one for connective K -theory in Section 4 and to the onein [15]. Here we only give an outline of the proof and the details are left to the readers. InΩ ∗ ( F l ∗ ( H p )), we have[ Y f ( w ) → F l ∗ ( H p )] = d Y i =1 c q i (ker( D i → D i − ) ⊗ E ∨ q i ) . (6.3)As before, the graded R -module homomorphism φ j : L R,j → Ω ∗ ( P ∗ (ker( H p j − → D j − )) isdefined by setting φ j ( t s · · · t s d d ) = τ s · · · τ s j − j − A [ j ] s j · · · A [ d ] s d . Then, similarly to Lemma 4.6and [15, Lemma 4.6], the definition of φ j together with (6.2) allows us to obtain π j ∗ ( τ s c q j (ker( D j → D j − ) ⊗ E ∨ q j )) = φ j t λ j j Y ≤ i We would like to thank David Anderson for his helpful commentson the earlier versions of the manuscript. The second author is supported by Grant-in-Aidfor Young Scientists (B) 16K17584. EXILLARY DEGENERACY LOCI CLASSES IN K-THEORY AND ALGEBRAIC COBORDISM 11 References [1] Anderson, D. K-theoretic Chern class formulas for vexillary degeneracy loci. Preprint.[2] Anderson, D., and Fulton, W. Chern class formulas for classical-type degeneracy loci. arXiv:1504.03615 .[3] Anderson, D., and Fulton, W. Degeneracy Loci, Pfaffians, and Vexillary Signed Permutations inTypes B, C, and D. arXiv:1210.2066 .[4] Buch, A. S. 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Thomas Hudson, Fachgruppe Mathematik und Informatik, Bergische Universit¨atWuppertal, Gaustrasse 20, 42119 Wuppertal, Germany email address : [email protected] Tomoo Matsumura, Department of Applied Mathematics, Okayama University ofScience, Okayama 700-0005, Japan email address ::